Motivated by recent studies of large Mallows(q) permutations, we propose a class of random permutations of N+ and of Z, called regenerative permutations. Many previous results of the limiting Mallows(q) permutations are recovered and extended. Three special examples: blocked permutations, p-shifted permutations and p-biased permutations are studied.
This paper offers some probabilistic and combinatorial insights into tree formulas for the Green function and hitting probabilities of Markov chains on a finite state space. These tree formulas are closely related to loop-erased random walks by Wilson's algorithm for random spanning trees, and to mixing times by the Markov chain tree theorem. Let mij be the mean first passage time from i to j for an irreducible chain with finite state space S and transition matrix (pij; i, j ∈ S). It is well known that mjj = 1/πj = Σ (1) /Σj, where π is the stationary distribution for the chain, Σj is the tree sum, over n n−2 trees t spanning S with root j and edges i → k directed towards j, of the tree product i→k∈t p ik , and Σ (1) := j∈S Σj. Chebotarev and Agaev [26] derived further results from Kirchhoff 's matrix tree theorem. We deduce that for i = j, mij = Σij/Σj, where Σij is the sum over the same set of n n−2 spanning trees of the same tree product as for Σj, except that in each product the factor p kj is omitted where k = k(i, j, t) is the last state before j in the path from i to j in t. It follows that Kemeny's constant j∈S mij/mjj equals Σ (2) /Σ (1) , where Σ (r) is the sum, over all forests f labeled by S with r directed trees, of the product of pij over edges i → j of f. We show that these results can be derived without appeal to the matrix tree theorem. A list of relevant literature is also reviewed.
E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron. AbstractThis paper is concerned with various aspects of the Slepian process (Bt+1 − Bt, t ≥ 0) derived from one-dimensional standard Brownian motion (Bt, t ≥ 0). In particular, we offer an analysis of the local structure of the Slepian zero set {t : Bt+1 = Bt}, including a path decomposition of the Slepian process for 0 ≤ t ≤ 1. We also establish the existence of a random time T ≥ 0 such that the processAs a natural candidate, the bridge-like process as below was considered:This bridge-like process bears some resemblance to Brownian bridge. At least, it starts and ends at 0, and is some part of a Brownian path in between. This leads us to the following question:The Slepian zeros and Brownian bridge embedded in Brownian motion Question 1.2.The study of the bridge-like process is challenging, because the random time F as in (1.2) does not fit into any of the above classes. We even do not know whether this bridge-like process is Markov, or whether it enjoys the semi-martingale property. Note that if the answer to (2) of Question 1.2 is positive, then we can apply Rost's filling scheme [17,72] as in Pitman and Tang [66, Section 3.5] to sample Brownian bridge from a sequence of i.i.d. bridge-like processes in Brownian motion by iteration of the construction (1.1). While we are unable to answer either of the above questions about the bridge-like process, we are able to settle Question 1.1.EJP 20 (2015), paper 61.
For the random interval partition of [0, 1] generated by the uniform stickbreaking scheme known as GEM(1), let u k be the probability that the first k intervals created by the stick-breaking scheme are also the first k intervals to be discovered in a process of uniform random sampling of points from [0, 1]. Then u k is a renewal sequence. We prove that u k is a rational linear combination of the real numbers 1, ζ(2), . . . , ζ(k) where ζ is the Riemann zeta function, and show that u k has limit 1/3 as k → ∞. Related results provide probabilistic interpretations of some multiple zeta values in terms of a Markov chain derived from the interval partition. This Markov chain has the structure of a weak record chain. Similar results are given for the GEM(θ) model, with beta(1, θ) instead of uniform stick-breaking factors, and for another more algebraic derivation of renewal sequences from the Riemann zeta function.
We study the convergence rate of continuous-time simulated annealing (Xt; t ≥ 0) and its discretization (x k ; k = 0, 1, . . .) for approximating the global optimum of a given function f . We prove that the tail probability P(f (Xt) > min f + δ) (resp. P(f (x k ) > min f +δ)) decays polynomial in time (resp. in cumulative step size), and provide an explicit rate as a function of the model parameters. Our argument applies the recent development on functional inequalities for the Gibbs measure at low temperatures -the Eyring-Kramers law. In the discrete setting, we obtain a condition on the step size to ensure the convergence.
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